Prep the chicken: Pat chicken dry with paper towels. If using breasts, slice large ones in half horizontally for even cooking.
Season both sides with 1 teaspoon salt, 1/2 teaspoon pepper, paprika, and Italian seasoning.
Mix the garlic Parmesan coating: In a small bowl, combine melted butter, olive oil, minced garlic, half the Parmesan (1/4 cup), and a pinch of red pepper flakes if using. Stir until glossy.
Preheat and heat the pan: Preheat the oven to 400°F (200°C) if finishing in the oven. Heat a large oven-safe skillet over medium-high.
Add a drizzle of olive oil.
Sear the chicken: Add chicken in a single layer. Sear 3–4 minutes per side until golden. It doesn’t need to be fully cooked yet.
Transfer to a plate.
Cook the green beans: Add green beans to the same skillet with a pinch of salt and a splash more oil if needed. Sauté 3–4 minutes until bright green and slightly tender but still crisp.
Add flavor to the beans: Spoon about 1–2 tablespoons of the garlic Parmesan mixture over the beans and toss to coat. Squeeze in a little lemon juice if you like.
Return chicken to the pan: Nestle the chicken back in among the beans.
Brush or spoon the remaining garlic Parmesan mixture over the chicken pieces.
Roast to finish: Sprinkle the remaining 1/4 cup Parmesan over everything. Transfer the skillet to the oven and roast 8–10 minutes, or until the chicken reaches 165°F (74°C) in the thickest part.
Add zest and rest: Remove from the oven, grate a little lemon zest over the top, and let rest 3 minutes. The zest brightens the richness.
Garnish and serve: Scatter chopped parsley over the skillet.
Taste and adjust salt or lemon. Serve hot with extra Parmesan at the table.